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WEAD 

Contributions  to  the 
History  of  Musical  Scales 


i-.  I.  krvi/j  i^i 


SMITHSONIAN    INSTITUTION. 

UNITED     STATES     NATIONAL     MUSEUM. 


CONTRIBUTIONS   TO    THE    HISTORY   OF 
MUSICAL   SCALES. 


BY 


CHARLES   KASSON  WPJAD, 

Examiner,  U.  S.  Fatent  Office. 


From  the  Report  of  the  United  States  National  JIuseum  for  1900,  jiages  417-463, 
with  teu  plates. 


WASHINGTON: 

GOVERNMENT    PRINTING    OFFICE. 
1902. 


1 


SMITHSONIAN    INSTITUTION. 

UNITED     STATES     NATIONAL     MUSEUM. 


CONTRIBUTIONS   TO    THE    HISTORY   OF 
MUSICAL   SCALES. 


CHARLES  KASSON  WEAD, 

// 

Examiner,  U.  S.  Patent  Office. 


From  the  Report  of  the  United  States  National  Museum  for  1900,  pages  417-463, 
with  ten  plates. 


WTW 


per\ 


^es'^ 


yORB 


WASHINGTON: 

GOVERNMENT    PRINTING    OFFICE, 
1902. 


^H  !c!^hr7 


CONTRIBUTIONS  TO  THE  HISTORY  OF 
MUSICAL  SCALES. 


CHARLES    KASSO?^^  WEAD, 

Examiner,  V.  S.  Patent  Office. 


417 


TABLE  OF  CONTENTS. 


■Page 

I.  Introduction 421 

II.  Stringed  instruments 424 

III.  Instruments  of  the  flute  type 426 

IV.  Instruments  of  the  resonator  type. -  -  428 

V.  The  influence  of  the  hand 433 

VI.  Composite  instruments - 436 

VII.  Conclusions - 437 

Appendix 441 


LIST  OF  ILLUSTRATIONS. 


PLATES. 

Facing  page. 

1.  Stringed  instruments 444 

2.  Flutes  with  equal-spaced  holes -  -  -  446 

8.  Flutes  with  equal-spaced  holes 448 

4.  Flutes  with  equal-spaced  holes 450 

5.  Flutes  with  holes  in  two  groups - -  452 

6.  Flutes  with  holes  in  two  groups 454 

7.  Central  American  resonators  or  whistles 456 

8.  Composite  instruments 458 

9.  Pan's  pipes 460 

10.  Scales  given  by  resonators 462 

TEXT  FIGURES. 

Page. 

1 .  European  mandolin,  after  VioUet  le  Due 424 

2.  Gi-eek  guitar,  after  Drieberg 424 

3.  Terra  cotta  whistle,  after  INIahillon 430 

4.  Babylonian  whistle,  after  Engel 431 

5.  Chinese  resonators,  after  Amiot 4^1 

6.  Globular  whistles,  after  Frobenius 432 

7.  Globular  whistle,  after  Kraus "^^^ 

8.  Xylophones,  after  Kraus ^^" 

419 


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CONTRIBUTIONS  TO  THE  HISTORY  OF  MUSICAL  SCALES. 


B\  Charles  Kasson  Wead, 
Examiner,  United  States  Patent  Office. 


I.   INTRODUCTION. 

In  the  development  of  musical  scales  four  stages  may  be  recognized: 

1.  The  stage  of  primitive  music,  where  there  is  no  more  indication 
of  a  scale  than  in  the  sounds  of  birds,  animals,  or  of  nature.  Students 
of  the  origin  of  music  may  give  free  rein  to  their  fancy  in  this  period, 
and  the  uncertain  musical  utterances  of  living  primitive  peoples  may 
be  construed  in  accordance  with  almost  any  prepossession  of  the  hearer. 

2.  The  stage  of  instruments  mechanicalh'  capable  of  furnishing  a 
scale.  This  stage  has  been  almost  entirely  overlooked  by  students  and 
is  the  special  subject  of  the  following  paper. 

3.  The  stage  of  theoretical  melodic  scales — Greek,  Arab,  Chinese, 
Hindu,  MediaBval,  etc.  All  the  original  treatises  concerning  these 
scales  imply  that  a  stage  of  development  has  been  reached  far  in 
advance  of  the  second.  Thousands  of  pages  have  been  written  on  this 
stage,  largeh^  polemical  and  lacking  in  insight,  for  the  subject  has  been 
a  dark  one;  but  Ellis  and  Hipkins's  work  of  1885  has  thrown  a  flood 
of  light  on  it. 

4.  The  stage  of  the  modern  harmonic  scale  and  its  descendent,  the 
equally  tempered  scale,  which  are  alike  dependent  both  on  a  theory 
and  on  the  possibility  of  embodying  it  in  instr.uments.  The  relation 
of  this  scale  to  the  present  study  will  be  noticed  later. 

These  four  stages,  of  course,  overlap  even  in  the  same  locality;  they 
correspond  in  a  rough  way  to  the  recognized  four  culture  stages, 
namely:  the  savage,  barbarous,  civilized,  and  enlightened. 

At  the  outset  it  should  be  recognized  that  the  only  working  hypothe- 
sis the  physicist  can  use  is  that  of  the  instrumental  origin  of  scales. 
Helmholtz's  view  that  the  harmonics  in  the  voice  and  in  the  tones  of 
instruments  were  influential  in  settling  the  positions  of  the  notes  of 
our  scale  is  obviously  consistent  with  this  hypothesis;  and  his  opinion 
that  this  influence  acted  on  other  scales  need  not  be  wholly  rejected, 
though  some  of  his  historical  authorities  were  untrustworthy,  and 

421 


422  REPORT    OF    NATIONAL   MUSEUM,  1900. 

.sonic  of  liis  coincidences  between  other  scales  and  the  harmonic  scale 
can  bo  explained  in  other  and  .sini])kn-  ways.  Writers  less  careful  than 
Helinholtz  have  made  the  assumption  that  these  harmonics  and  the 
constitution  of  the  ear  must  have  guided  primitive  musicians  to  a  sub- 
stantially harmonic  scale;  and  one  writer  has  even  maintained  that 
instruments  corrupted  the  taste  of  men.  But  as  j'et  there  has  been 
no  such  l)ody  of  facts  collected  in  support  of  this  assumption  as  need 
delay  one  following  out  the  other  theory.  Of  course  the  knowledge 
of  the  scales  is  only  a  stepping-stone  to  the  understanding  of  the  music 
and  something  of  the  life  of  a  people;  so  some  day  the  materials  worked 
into  shape  by  the  physicist  may  b(»  built  into  a  fairer  structure  bj^  the 
psychologist. 

The  broad  fact  which  undeilies  all  study  of  scales  was  recognized  by 
the  (ii'(M>k  nmsician  Aristoxenus  three  centuries  ))efore  the  Christian 
era,  lie  pointed  out  that  the  voice,  in  speaking,  changes  its  pitch  by 
insensible  gradations,  while  in  singing  it  moves  mostly  by  leaps.  We 
recognizi^  the  same  fact  when  we  say  that  a  singer  follows  a  scal<\  but 
do  not  say  it  of  a  speaker.  The  one,  to  use  the  common  figure,  ascends 
or  descends  a  ladder  or  staircase;  the  other  follows  a  continuous  slope, 
and  may  never  step  twice  in  the  same  place.  Now,  it  is  quite  possible 
that  in  a  song  the  voice  may  always  move  by  leaps,  and  in  repeating 
the  song  always  take  the  same  leaps  as  closely'  as  can  be  observed,  yet 
never  strike  a  note  which  it  has  struck  before;  just  as  one  ma}'  toss 
a  stone  up  and  down  on  a  hillside,  marking  each  time  where  it  lands, 
and  after  a  hundred  tosses  finds  it  had  not  landed  twice  at  quite  the 
same  level,  or  in  striding  up  and  down  hill  may  never  plant  his  foot 
twice  at  the  same  level.  1  think  this  was  the  character  of  the  songs 
of  the  tirst  stage  and  of  much  primitive  song  to-day,  though  the 
evidence  is  too  scanty  to  l)e  conclusive. 

Iloweverthis  ma}'  be,  it  is  certain  that  most  peoples  wlio  have  attained 
any  moderate  degree  of  civilization  have  attempted  to  limit  the  num- 
l)er  of  steps  to  be  taken  by  the  voice  in  any  song  between  the  highest 
and  lowest  note,  and  to  Hx  these  steps  ))y  rules,  so  that  many  men 
may  learn  them  and  be  in  substantial  agreement.  Various  old  writers 
give  the  ruh  s  in  Aogue  among  (xreek  theorists;  in  the  last  century 
Amiot  described  the  Chinese  rules,  while  in  the  last  two  decades  the 
rules  of  Aral),  Hindu,  Japanese,  and  Siamese  musicians  have  been 
made  accessible.  The  most  familial'  rules,  as  is  well  known,  depend 
on  that  law  of  vibrating  strings  which  is  followed  by  a  violinist  in  his 
Hngering — namely,  that  the  frequency  of  vibration  of  parts  of  any 
stretched  string  is  inversely  as  the  length  of  the  parts,  provided  the 
tension  does  not  change.  Our  latest  rule,  historically  derived  from 
one  of  the  many  (ireek  and  Arab  rules  by  subdividing  the  whole  tones, 
so  giving  twelve  ste])s  to  the  octave,  is  embodied  on  the  neck  of  a  guitar 
or  inundolin:   here  it  is  obvious  that  the  successive  stopping  points  as 


HISTORY    OF    MUSICAL    SCALES  423 

marked  by  frets  get  closer  and  closer  together  as  the  pitch  rises.  All 
musicians  know  that  this  number  of  notes,  twelve,  is  found  confus- 
ingly great  for  ordinary  playing,  and  know  the  principles  ])y  which  the 
player  selects  certain  notes  for  any  tune.  But  this  nudtiplicity  of 
notes  has  an  iixiportant  bearing  on  all  studies  on  nonharmonic  nuisic 
made  Iw  harmonic  musicians.  For  ever}"  sound  within  the  compass 
of  the  instrument  comes  very  near  to  some  one  of  the  twelve  notes  and 
may  readily  be  represented  thereby,  owing  to  the  difficulty  the  hearer 
has  in  estimating  deviations  from  the  familiar  series  and  in  noting  them 
down.  The  results  of  this  approximation  are  to  mask  all  deviations 
from  the  twelve-tone  piano  scale,  whether  intentionally  or  accidentally 
made,  and  to  make  it  appear  to  musicians,  first,  that  nearly  all  the  music 
of  the  world  is  performed  substantially  in  our  scale;  and  second,  that 
any  other  theoretical  scales,  such  as  those  found  among  Orientals,  or 
described  b}^  our  European  ancestors,  are  merely  mathematical  jug- 
glery and  of  as  little  significance  as  proposals  for  a  change  that  occa- 
sionally appear  in  modern  musical  or  scientific  journals. 

It  is  the  purpose  of  this  paper  first  to  describe  several  types  and 
forms  of  instruments  widely  used,  each  embodying  a  principle  of  scale 
building  distinctly  unlike  ours,  though  sometimes  giving  a  result  that 
seems  surprisingly  familiar.  Nearl}"  all  these  instruments,  it  will  be 
noted,  belong  to  what  was  called  above  the  second  or  barbarous  stage, 
though  a  few  of  them  come  from  countries  where  musicians  have 
reached  the  third  and  fourth  stages.  A  second  purpose  is  to  present 
a  new  and  generic  principle  of  primitive  scale-building  applicable  to 
the  various  types  of  instruments  discussed. 

But  before  going  further  it  must  be  recognized  that  the  word 
'"scale"'  has  many  meanings.  Perhaps  the  lowest  and  loosest  is — the 
series  of  sounds  used  in  sk.ny  musical  performance,  arranged  in  order  of 
pitch.  The  one  that  will  most  closely  fit  the  present  needs  is — the  series 
of  sounds  produced  upon  a  particular  mstrument;  while  the  most 
exact  definition,  but  one  applicable  only  where  nuisical  principles  are 
well  developed  is  this: 

A  male  is  mi  md&pe7idently  repr<)d>Lcihle  series  of  sounds  arranged 
hi  order  of  2) itch ^  recognized  as  a  standard  and  iitted  for  musical 
purposes. 

While  the  last  two  definitions  imply  an  instrument  in  which  tne 
scales  are  embodied,  the  limitation  is  in  appearance  onl}",  for  there  is 
no  evidence  that  any  musicians  do  have  a  standard  series  of  tones, 
unless  they  have  one  or  more  instruments  embodying  it,  and  have 
learned  the  series  directly  or  indirectly  from  such  an  instrument. 


424 


REPORT    OF    NATIONAL    MUSEUM,   1900. 


II.   STRINGED   INSTRUMENTS. 


'^1 


111  sharp  coiitrust  to  that  widely  used  division  of  a  string  which  we 
know  on  the  guitar,  showing  decreasing  distances  between  the  frets  as 
the  pitch  rises,  wo  tind  many  instances  of  a  uniform  spacing  of  the 
frets  tlirough  a  considerabhi  distance.  Instances  from  four  countries 
may  here  be  cited: 

1.  The  well-known  architect,  VioUet-lc-nuc.'  gives  a  figure  (fig.  1) 
of  a  Miaiidoliji  from  the  end  of  the  sixteenth  century  which  shows  frets 
for  the  first  seven  semitones  pretty  uniformly 
spaced;  the  frets  for  the  next  five  to  complete  the 
octave  are  again  imiform,  though  closer  than  be- 
fore, and  the  following  five  are  also  uniformly 
spaced  and  still  closer.  Figures  in  other  books  ^ 
of  European  lutes,  viols,  etc.,  ver}'  often  show  a 
similar  equal  spacing.  These  are  too  numerous 
to  be  lightly  treated  as  artists'  blunders.  Two 
instruments  in  the  United  States  National  Museum 
are  illustrated  in  Plate  1. 

"I.  Among  the  Greek  rules  given 
by  Ptolemy  is  one  for  the  division 
called  Dlatonon  homalon^  in  which 
the  whole  string  being  twelve  units 
long  the  points  for  stopping  would 
be  at  11,  10,  9,  and  8,  giving  C,  a 
note  between  Dt>  and  D,  Et>,  F,  and 
G.  Here  it  will  be  noticed  the  inter- 
vals get  larger  and  larger  as  the  pitch 
rises.  Again,  Carl  EngeP  refers  to 
Drieberg's  drawing  of  the  ancient 
Greek  guitar  in  the  Berlin  Museum, 
which  has  "seven  frets  at  equal  dis- 
tances," but  objects  to  it  as  it  does 
not  give  a  diatonic  scale.  The  tracing  of  this  drawing 
fui-nished  l)v  Professor  Howard,  of  Harvard,  adds  to 
Engel's  data  the  fact  that  the  whole  compass  of  the  six 
intervals  is  slightly  more  than  an  octave  (fig.  2). 

3.  Among  the  instruments  described  in  the  Arabic  treatise  of  the 


Fig.l. 


EUROPEAN  M.VNDOLIN. 
After  VtoIlet-le-Duc. 


Fig.  2. 
GREEK  GUITAR. 
After  Drieherg:. 


Plates  V,  fig.  ;^;  vi, 
Rome,  1776.     Plates 


'  Dictionnaire  raiwinne  du  mobilier  fran^ais,  II,  1S71,  ]>!.  li. 
■■'M.  Pnetorius,  Syntafrina  Municum,  II,  IfilS.     Reprint,  1894 
fig.  1;  XVI,  fig.  1;  XVII,  fig.  4;  xx,  figs.  1,  H. 

Bonaiini,  Deseription  des  instruments  hannouiqnes.    2(1.  cd. 

LII,  LVII,  LX,   LXXI. 

J.  Rulilinann,  (U'schiulitc  dcr  Bogeninstrutnonto,  1882.     I'latcs  ix,  figs.  2,  o,  6,  18; 
X,  fig.  Ri;  XIII,  figs.  8,  8. 

'Musif  of  the  Most  Aiu-ient  Nations,  18(14,  [..  2()r>. 


HISTORY    OF    MUSICAL    SCALES.  425 

famous  Al  Farabi,^  who  died  960  A.  D.,  is  the  short-necked  tanhour 
of  Bao'dad,  usually  having-  two  strings:  on  this  a  fret  was  first  placed 
at  one-eighth  the  length  of  the  string  from  the  upper  end,  and  this 
space  then  divided  into  five  equal  parts.  As  the  compass  on  each  string 
was  but  little  over  a  whole  tone,  each  step  was  about  a  quarter-tone. 
These  ligatures  or  frets  are  called  ' '  heathen  "  or  "  pagan,"  and  the  tunes 
played  on  them  "heathen  airs,"  clearl_y  indicating  that  there  was  a 
scale  native  to  the  people  whom  the  Mohammedan  armies  had  con- 
quered, a  scale  utterly  different  from  either  that  of  the  lute  or  the 
tanbour  of  Khorassan,  with  their  resemblances  to  Greek  scales.  Three 
hundred  years  later,  or  about  1250  A.  D.,  Safi-ed-din,*^  a  famous  musi- 
cian of  Bagdad,  wrote  for  his  pupil,  the  son  of  the  Vizier,  a  Treatise 
on  Musical  Ratios.  He  based  them  on  string  lengths,  and  in  discussing 
instruments  gives  a  figure  of  the  frets  on  the  neck  of  the  lute,  and  it  is 
noteworthy  that  these  are  equally  spaced  over  a  distance  of  a  quarter 
length  of  the  string.  Further,  he  explains  how  of  the  ten  frets  in  this 
short  distance,  located  by  various  rules,  five  were  fixed  by  arithmetical 
bisection  or  halving  of  the  space  between  two  frets  already  fixed;  one 
of  these,  midway  between  what  we  should  call  D  and  E,  if  the  open 
string  gives  C,  was  called  the  "Persian  middle,"  and  was  very  much 
in  use  in  his  time.  Safi-ed-din''  further  describes,  in  two  connections, 
a  division  of  the  Fourth,  like  the  Greek  one  already  quoted,  where  the 
string  lengths  are  12,  11,  10,  9,  saying-  it  is  consonant  and  much  used; 
in  fact  it  is  preferred  to  one  that  is  substantially  like  the  theoretical 
diatonic  scale;  still  it  should  be  added  that  when  he  comes  to  arrange 
intervals  to  make  up  two  octaves  he  puts  our  arrangement  along  with 
the  most  agreeable  half  dozen  genera. 

4.  In  India  there  has  been  in  modern  times  a  curious  reversion  from 
an  elaborate  historical  scale  of  twenty-two  steps  to  the  octave,  of  which 
no  modern  Hindu  or  European  knows  the  theory,  to  an  equal  linear 
division;*  one-half  of  the  string  on  the  sito/r  is  bisected;  the  first  or 
end  quarter-length  is  then  divided  into  nine  parts,  each  marked  by  a 
fret,  and  the  second  quarter-length  into  thirteen  parts  similarly 
marked.  Out  of  the  twentj^-three  tones  within  the  octave  the  player 
selects  a  limited  number,  five,  six,  or  seven,  rarely  eight,  for  any  par- 
ticular tune.  Most  of  the  notes  used  are  found  on  calculation  to  be 
deceptively  close  to  the  notes  of  our  chromatic  scale,  and  so  may  be 
easily  confounded  with  them  by  European  hearers. 

5.  This   arithmetical   division   has    been    advocated    by    European 

'  Land's  translation  in  Travaux  de  la  6^  session  du  Congres  Internationale  des  Orien- 
talistes  ii  Leide,  1883,  pp.  107-114. 

^Carra  de  Vaux's  translation  in  Journal  Asiatique,  XVIII,  1891,  p.  330. 

^Idem,  pp.  308-317. 

*Tagore,  Musical  Scales  of  the  Hindus,  Calcutta,  1884,  supplement.  Partly  quoted 
by  C.  R.  Day,  Music  ...  of  Southern  India,  1891,  and  Ellis,  Journal  Society  of  Arts, 
XXXIII,  1885,  p.  502. 

NAT  MUS   1900- 30 


420  RErOKT    OF    NATIONAL    MUSEUM,  IWO. 

tlit'orists,  as  l»v  Jiiiiuiid'  in  u  troati.sc  of  IT.MI.  a  C()i>y  of  wliich  is  in  the 
Lniox  Lil»raiv.  Nt'w  Voik:  and  Ft'tis  '  in  his  brief  account  of  this 
author  refers  to  otiicrs  who  niaintaincd  siniihir  views. 

III.    INSTRUMENTS    OF    THE    FLUTE    TYPE. 

The  simple  lhitr>  arc  instruments  of  a  type  more  primitive  and  more 
widelv  distrilmted  than  fretted  strino-ed  instruments.  These  instru- 
ments are  sometinu's  side-hlown,  as  is  the  case  with  the  modern  flute; 
».!•  end  hh)wn,  as  one  ]>K)Ws  into  a  key  or  pan's  pipe;  or  blown  with  a 
whisth"  mouthpiece,  like  the  flageolet;  or  blown  w'ith  a  weak  reed,  as 
the  oboe.  For  the  purposes  of  this  discussion  the  mode  of  excitinj^ 
the  vibrtition  is  inmiaterial.  All  of  them  embody  the  law^  that  the 
frtMiueiiiv  of  vibration  of  a  column  of  air  in  a  tube  depends  mainl3'on  its 
len<rth.  and  the  variation  in  leng-th  of  the  air  column  so  as  to  produce 
several  sounds  fi'om  one  tube  is  produced  b}'  opening  holes  in  the 
sides  of  the  tul>e.  in  i)ractice  these  holes  never  can  open  so  freely  to 
the  outside  air  that  the  portion  of  the  tube  beyond  them  may  be  con- 
sidered as  removed  (the  po.ssibility  or  necessit}'  of  cross-tingering 
proves  this  to  the  player),  so  the  proper  location  and  diameter  of  the 
holes  to  ])roduce  the  notes  of  our  scale  of  even  quality  are  fixed,  not 
by  a  simple  law  as  the  frets  on  the  guitar  are  located,  but  by  laborious 
exj)erimenting  to  get  a  standard  instrument  which  is  then  reproduced 
with  Chinese  fidelity. 

Now.  as  one  looks  over  a  collection  of  wind  instruments,  like  the 
splendid  one  in  the  U.  S.  National  Museum,  or  examines  flutes  figured 
in  books,  it  will  be  easy  to  recognize  that  there  are  two  principal 
types  (A)  those  having  the  holes  spaced  at  sensibly  equal  distances, 
and  (B)  those  having  two  groups  each  of  three  equally  spaced  holes, 
tlie  interval  l)etween  the  nearest  holes  of  the  two  groups  being  obvi- 
ously orcater  than  that  between  the  holes  of  each  group.  As  the 
common  primitive  method  of  making  the  holes  is  by  burning,  the 
holes  are  generally  more  uniform  in  diameter  than  those  on  European 
rtutes  of  a  century  aiLjo. 

Illustrations  of  flutes  of  type  A  are  found  in  Engel's  Musical 
Instruments,  some  of  which  are  copied  on  Plate  ri.  Dr.  Wilson's 
paper  (»n  Trehistctric  Art '  has  many  more  illustrations,  as  the  figures  of 
bone  tlutes  from  ('ost;i  Uica  and  British  (hiiana.  of  pottery  flutes  from 
Mexico  and  the  Zufii  Indians,  of  tubes  with  a  simple  reed  from  Egypt 
and  Palestine,  of  wood(Mi  llutes  brought  from  Thibet  by  Mr.  Rockhill, 
and  a  wooden  Mute  from  the  Kiowa  Indians.  Fetis*  has  a  cut  of  the 
staghorn  flute  from  the  stone  age  witli  tlir(>e  equidistant  holes,  referred 


'. laniard.  Ucclicnhos  snr  la  Tluorie  de  la  Musique. 
'■"Frtis,  I?i(»);rapliii-  ^^nivl■r^^^■lll■  (U-.s  Music-ienis. 
'l{.-lM)rt  Mf  the  V.  S.  National  Mnseiini  for  1896,  i)p.  :)25-664. 
*  Ilistoiri-  gc'iu'ralf  ik-  la  imisiqin.-,  I,  j..  26. 


HISTOKY    OF    MUSICAL    SCALES.  427 

to  1\Y  Wilson  (p.  526).  So  far  as  is  Iviiown  not  one  of  the  peoples 
from  whom  these  instruments  have  come  has  any  musical  theory,  but 
some  of  them  do  have  a  principle  of  instrument  construction;  for  a 
partly  educated  young-  Kiowa  Indian,  in  Washington  a  few  years  ago, 
in  a  party  under  charge  of  Mr.  James  Mooney,  showed  the  writer  how 
the  holes  on  a  flute  on  which  he  placed  were  located  by  measuring- 
three  ling'er-breadths  from  the  lower  end  to  the  lower  hole,  and  then 
taking-  shorter  but  equal  spaces  for  the  succeeding-  holes.  The  inter- 
preter added  that  he  had  seen  the  holes  spaced  by  cutting  a  short  stick 
as  a  measure.  The  late  Mr.  F.  H.  Gushing  has  furnished  the  addi- 
tional fact  that  measurement  by  finger-breadths  is  very  common  among 
Indians;  and  Dr.  Fewkes^  gives  a  figure  to  show  how  the  prayer 
sticks,  used  by  the  Hopi  Indians  in  the  Snake  ceremonials  at  Walpi, 
are  measured  oft'  into  seven  parts  by  the  distances  from  creases  on  the 
hand  to  the  tip  of  the  finger.  On  the  Kiowa  flute  (Plate  4,  No.  2)  the 
distance  between  the  centers  of  the  holes  is  32  mm.,  which  is  two 
medium  finger-breadths.  Some  instruments  of  this  type  belonging  to 
the  U.  S.  National  Museum  are  show^n  in  Plates  3  and  4. 

But  it  is  not  only  among  primitive  and  prehistoric  peoples  that 
such  a  succession  of  holes  is  found.  The  common  military  fife  has  it. 
The  bagpiper  recently  seen  on  the  streets  of  Washington  used  a 
chaunter  (oboe),  the  holes  of  which  were  at  sensibly  equal  distances, 
so  conforming  to  the  well-known  fact  that  the  bagpipe  scale  is  inten- 
tionally unlike  the  harp  scale.  A  Japanese  Fouye  with  7  holes  figured 
in  the  catalogue  of  the  Kraus  collection  at  Florence  shows  to  the 
eye  holes  at  nearly  equal  spaces,  and  has,  as  reported,  the  steps  of 
the  scale  increasing  in  length  as  the  pitch  rises.  From  Egypt'  there 
have  come  twenty-five  3-  and  4-hole  ancient  flutes,  or  more  exactly, 
oboes,  and  a  few  of  5,  6,  and  more  holes.  One  of  the  4-holed  instru- 
ments from  a  tomb  of  about  1100  B.  C.  shows  the  holes  35  mm.  apart 
and  the  lowest  hole  twice  this  distance  from  the  bottom.  Villoteau's^ 
plates  of  modern  Egyptian  instruments  show  various  types  of  tubes 
with  equally  spaced  holes. 

Flutes  of  the  second  or  B  type  with  two  groups  of  equal-spaced 
holes  were  sold  in  quantities  at  the  Java  village  at  the  World's  Fair 
held  in  Chicago  in  1893  (Plate  6,  No.  1).  No  two  of  the  instruments 
seemed  to  have  the  same  length  or  location  of  holes,  but  this  group- 
ing was  unmistakable.  Of  this  type  is  also  a  curious  ancient  Chinese 
instrument,  the  Tche,  described  by  Amiot,*  closed  at  both  ends  with 


^Journal  of  American  Ethnology  and  Archaeology,  IV,  1894,  p,  25-26. 

^Loret,  Journal  Asiatique,  8th  ser.,  XIV,  1889,  pp.  Ill,  197.  Musical  Times,  Lon- 
don, XXXI,  1890,  pp.  585,  713. 

^Description  de  1' Egypt,  Etat  moderne,  II,  1809,  plate  cc. 

*Memoires  concernant  I'histoire  ....  des  Chinois,  VI,  1780,  p.  76,  pi.  vi,  fig.  42. 
Mahillon,  Brussels  Conservatory  Catalogue  I,  No.  865. 


42K  KKI'oKI     OK    NATIONAL    MTSKUM,   1900. 

ill)  riiibouclifrr  at  the  iniddlc  and  holes  syimiieti-ically  placed  on  each 
sidj'  dividintr  the  whole  leiij^th  into  thirds,  (luarter*^,  and  sixths;  so,  if 
the  whole  lenj^-th  is  culled  12,  the  mouth  hole  is  at  (5  and  the  finger 
holes  at  '2,  H,  4,  S. '.>.  and  10.  Mahillon  coi)ied  the  instrument.  I)utdi<l 
not  close  the  ends,  and  reports  the  scale  as  a  chiomatic  one  from  Eto  A#. 
Most  of  the  old  European  wood  wind  instruments  tigured  by  Prieto- 
rius'  (l»)ls)  an>  conspicuously  of  this  type,  as  the  appended  Plate  5 
shows  without  necessity  of  description,  and  various  similar  instru- 
ments of  the  Museum  collections  are  figured  in  Plate  0. 

IV.  INSTRUMENTS  OF  THE  RESONATOR  TYPE. 

1.  The  next  iiToup  includes  a  variety  of  instruments  of  the  resonator 
tvpe.  a  type  that  is  widely  distrit)uted  and  conforms  to  a  law  hitherto 
imrecogni/ed  as  capable  of  furnishing  a  scale;  though  Sondhaus  in 
1850  .stilted  the  law  and  tried  a  few^  rough  experiments.  The  mathema- 
ticians- have  pro\ed  that  a  mass  of  air  in  a  confined  space  with  a  ver}^ 
small  nearly  circular  opening,  as  a  short-necked  bottle  or  a  whistle, 
has  a  fre((uency  of  vibration  proportional  to  the  square  root  of  the 
fraction  which  expresses  the  diameter  of  the  hole  divided  by  the  vohmie 
of  the  cavity;  and  if  there  are  two  such  openings  so  placed  that  the 
tiow  of  air  through  one  does  not  interfere  with  that  through  the  other, 
the  numerator  of  the  fraction  will  be  the  sum  of  the  two  diameters.  Now 
t\xtend  the  same  principle,  and  one  may  have  a  series  of  sounds  rising 
in  pitch  as  one  after  another  of  several  holes  in  the  wall  is  opened; 
and  pi'ovided  the  chai-actei'  of  the  vibration  is  not  essentially  changed, 
the  fretpuMUT  of  vibration  of  these  notes  will  increase  as  the  square 
root  of  the  sum  of  the  diameters  of  the  holes  opened.  Suppose,  for 
example,  that  a  vessel  has  one  mouth-hole  of  diameter  2  and  several 
properly  placed  finger-holes  of  diameter  1;  then  on  successivelv  open- 
ing these  a  scale  mav  be  produced  having  vibration  frequencies  in  the 
ratio  of  the  scjuare  roots  of  2,  3,  4,  5,  etc.  A  moment's  consideration  will 
show  that  in  such  a  scale  the  intervals  betw'een  successive  sounds 
betome  hss  iind  less  as  the  pitch  rises,  instead  of  becoming  greater  as 
is  the  ca.-e  with  sti'ings  oi-  fiutes  where  the  spacing  of  frets  or  holes  Wj 
unifoiin. 

I  lie  niii-i  elaborate  and  beautiful  illusti'ations  of  instruments  of  this 
tvpe  are  from  graves  in  ('(Mitral  and  South  America.  (See  Plate  T.) 
1  he  Pnited  States  National  Museum  has  many  whistles  from  Chiricpii 
in  ("olniiibja.  m<)>t  of  them  giving  but  a  single  high  note;  these  differ 
sub>tantially.  it  will  be  noticed,  from  stopped  organ  pipes,  since  in  the 
latter  the  mouth  extends  the  full  width  of  the  tube.  Whistles  with 
one  or  two  linger-holes  have  come  from  Mexico  and  San  Salvador,  but 
the  niM.t  complet(>  and  pei-fect  are  fiom  Costa  Rica.     Of  these  the  one 


'l^yiitayiiiii  Mu.sicuni,  ]»ls.  i.\  and  x. 

'liayli-iKli,  Theory  (if  Sunnil,  II,  1878,  Chap.  xvi. 


History  of  musical  scales.  429 

bearing-  the  catalogue  number  59970  (Plate  T,  lig.  1)  has  served  as  the 
type  specimen,  and  is  the  instrument  which  led  to  this  investigation. 
It  has  a  globular  body  with  bird's  head,  a  mouthpiece  about  in  the 
position  of  a  bird's  tail,  and  four  finger-holes  on  the  ])ack  symmetri- 
cally placed;  these  holes  seem  to  be  precisely  equal  in  diameter,  and 
equivalent  in  nuisical  effect,  so  the  order  of  fingering  is  a  matter  of 
indifference,  and  all  the  tones  are  clear  and  distinct;  in  Dr.  Wilson's 
paper, ^  Mr,  Upham,  who  is  a  violinist,  notes  them  as  F,  A,  C,  D,  E, 
On  measurement  the  volume  wa^;  found  to  be  36.0  cc,  the  equivalent 
diameter  of  the  trapezoidal  mouth  hole  1  cm.,  and  the  diameter  of  the 
hnger  holes  .65  cm.;  these  diameters,  however,  need  a  correction  on 
account  of  the  thickness  of  the  walls,  since  the  air  can  not  pass  freely 
through  the  rather  thick  wall.  The  final  result  of  the  calculation  is 
to  give,  with  all  finger-holes  closed,  the  note  F  on  the  highest  line  of 
the  treble  staff',  to  within  half  a  semitone,  and  on  opening  the  finger- 
holes  in  any  order  to  give  the  succession  of  intervals  4,  3,  2,  and  2 
equal  semitones,  with  a  mean  error  of  only  one-eighth  E.  S.  Accord- 
ing to  the  theory  the  series  of  intervals  depends  only  on  the  ratio 
between  the  diameters  of  the  holes  and  the  mouth  hole,  in  this  case  1 
to  1.62;  so  the  series  of  tones  has  vibration  frequencies  approximately 
as  the  square  roots  of  1.6,  2.6,  3.6,  4.6,  5.6,  or  of  1,  1.62,  2.24,  2.86, 
3.48;  but  the  pitch  of  all  depends  on  the  quotient  of  the  radius  of  the 
mouth-hole  by  the  volume.  Although  the  theoretical  correction  for 
thickness  of  wall  can  not  be  quite  precise,  it  affects  all  the  holes  to 
nearly  the  same  extent,  and  the  greatest  probable  error  that  can  be 
assumed  will  not  change  the  whole  compass  more  than  half  a  semitone; 
so  the  calculated  scale  would  still  be  substantially  what  the  ear  con- 
firms— F,  A,  C,  D,  E,  or  in  syllables  d^>^  mi,  sol,  la,  si. 

The  Museum  has  several  other  Costa  Rican  instruments  also  of 
pottery  quite  similar  in  appearance  to  this,  but  not  capable  of  giv- 
ing such  clear  tones,  or  quite  so  perfect  in  the  equality  of  the  holes. 
If  the  holes  are  unequal  in  diameter,  in  thickness  of  wall,  or  in  loca- 
tion  with  reference  to  the  vibrating  mass  of  air,  the  order  of  pitch  will 
depend  on  which  holes  are  opened  instead  of  merel}^  on  how  many; 
with  five  holes  sixteen  combinations  are  possible;  but  of  the  eleven 
instruments  in  the  Museum  eight  give  only  five  notes  each,  two  give 
seven  notes,  and  one  gives  nine  notes.  If  the  finger-holes  are  small 
relatively  to  the  mouth  hole,  the  compass  is  small,  so  one  high-pitched 
whistle  has  a  compass  of  only  six  semitones — G  to  C# — and  another  runs 
from  B  to  E;  three  have  a  compass  of  seven  E.  S. ,  that  is,  a  musical 
fifth,  and  two  each  have,  respectively,  eight,  nine,  and  eleven  semitones. 

Still  other  National  Museum  instruments,  similar  in  principle,  but 
ruder  in  workmanship  and  more  grotesque  in  form,  have  come  from 
Chiriqui,  Columbia,  and  are  figured  in  Dr.  Wilson's  report,  pages  628 

1  Report  of  United  States  National  Museum  for  1896,  p.  617. 


41M) 


REPORT    OF    NATIONAL    MUSET^M,    ]9(»0. 


to  ♦i4«>.  In  other  iiiiix'iiiiis  similar  iiistiumciits  are  to  he  found.  A 
few  from  ('liiii(|ui  were  Itriefly  descri])od  forty  years  ago  as  belonging 
to  the  American  Kthiioh)ei(al  Soeiety.' 

In  the  American  Museum  of  Natural  History  in  New  York,  as 
reported  l)v  Prof.  V.  W.  Putnam,  half  a  dozen  such  three- and  four- 
iiole  wiiistles  from  the  region  of  Santa  Marta.  Colombia,  are  to  })e 
.seen:  wiiile  undei-  his  charge  at  Cambridge,  Ma.ss,,  there  are  a  number 
from  the  I'loa  \alley.  Central  America*/  of  those  figured,  three  have 
thive  tinger-holes  and  are  said  to  give  five  notes  each. 

In  the  Brussels  Con.servatory  Collection''  there  are  twenty-rive  terra 
eotta  instruments  from  Mexico;  tw^o  of  them  are  clearly  of  this 
resonator  tvi)e,  giving  rive  notes  and  having  a  compass,  respectively, 
of  eight  and  eleven  E.  S.  (rig.  3).  Lastly,  a  similar 
instrument  described  and  rigured  ])y  Dr.  Walter 
Hough,  in  the  Report  on  the  Columl)ian  Historical 
Exposition  at  Madrid.  l.SH2-18y3,  has  the  small  com- 
pass of  six  E.  S.  The  point  should  again  be  empha- 
size^l  that  with  th("se  instruments  the  notes  get  closer 
and  closer  together  as  the  pitch  rises;  for  instance,  on 
the  type  instrument  the  successive  intervals  are  in 
whole  numbers4, 3, 2. 2,  E.  S. ;  on  the  Brussels  instru- 
ments, 3,  2,  2, 1,  and  4,  3,  2,  2;  on  the  Madrid  speci- 
men. 2,  2,  1,  1.  A  chart  (Plate  10)  will  show  more, 
accurately  what  th(»  four  intcn'vals  are  with  any  speci- 
fied ratio  of  holes,  and  whether  there  is  appreciable 
error  in  expressing  the  interval  in  whole  numbers. 
Of  course  the  calculations  assume  uniformity  in  the 
l)lo\\  iiig.  for  it  is  easy  for  the  performer  to  vary  the  notes  by  a  con- 
si(leiat)le  amount.  .Still,  it  is  a  surprise  to  rind  how  well  the.se  simple 
.seaU's  satisfy  th(^  ear. 

A  sort  of  stone  riageolet  from  Costa  Rica  appears  to  be  connected 
with  the.se  instruments  in  principle  (Plate  7,  rig.  8).  This  is  closed 
;J  one  end  and  has  a  small  mouth  opening  and  four  ringer-holes 
arrangefl  in  pairs;  its  scale  of  seviMi  notes  from  rive  holes  proves  that 
the  holes  are  not  acoustically  etiuivalent.  t»ut  the  two  of  each  pair  are 
found  to  l»e  nearly  e(juivalent;  so  on  trial  it  apjiears  that  the  s(|uare 
root  foiimda  may  l)e  ajjplied.  by  giving  to  the  mouth-hole  the  value  5, 
to  rath  of  the  iiejii'er  holes  the  \alue  l.and  to  the  other  holes  the  value 
'2:  then  the  vibration  fre(|uencies  will  be  as  the  square  roots  of  the 
numbers  .")  to  11.  The  calculated  intervals  from  the  lowest  note  are 
l.t;.  2.'.t.  4.1,  T).  1.  t;.(».  t;.s  E.  S. :  the  ob.siwved  intervals  are  2,  3,  4,  5, 
t),  and  7  E.  S. 


Fig.  -S. 

TERRA  COTTA   WHISTLE 

AftiT.Mnhillon. 


'  MiUra/itie  of  American  History,  IV,  1860,  pp.  144, 177,  240,  274. 
•Memoirs  of  tl)e  rcaliody  .Museum,  1,  No.  4,  pi.  ix. 
'  .Maliilloii'>4  Catahiguf,  II,  Nos.  852,  853. 


HISTORY    OF    MUSICAL    SCALES. 


431 


BABYLONIAN  WHISTLE 
Attei-  EngeL 


2.  A  pottery  whistle  found  in  the  ruins  of  Babylon,  dating  probably 
from  about  500  B.  C,  is  in  the  Museum  of  the  Royal  Asiatic  Society, 
London^  (tig.  4).  Rowbotham^  says  this  is  similar  to  the  reindeer 
joint  used  by  the  cave  men.  Its  extreme  length  is  8  inches  and  it 
has  two  finger  holes.  The  three  notes  are  stated  to  be  C  (of  525  d.  v.), 
E,  and  G:  but  the  holes  not  being  quite 

equal,  the  Pv  from  one  of  them  is  a  quarter  of 
a  tone  flat.  B}'  blowing  hard  the  G  can  be 
carried  up  to  A.  The  chart  (Plate  10)  shows 
that  if  the  interval  C-G  is  exact,  with  equal 
holes  the  intermediate  note  E  will  be  a  very 
little  sharp  of  the  piano  note,  but  the  differ- 
ence is  only  about  1  per  cent,  one-fifth  of 
a  semitone,  and  so  is  utterly  negligible  in 
notes  of  such  uncertain  intonation. 

3.  Striking  comparisons  have  sometimes 
been  made,  and  especially  by  the  late  Prof. 
Terrien  de  la  Couperie,  between  the  Assvrian 
and  earh^  Chinese  civilizations.  Whatever 
their  relations  may  have  been,  it  is  curious 
that  the  only  instrument  of  the  resonator 
type,  having  several  finger  holes  and  coming 
from  a  people  who  had  a  musical  theory,  is 
the  Hsilan  (Van  Aalst)''  or  Hiuen  (Amiot)^  of  the  Chinese,  said  to  have 
been  invented  some  2,700  years  before  our  era,  and  still  used  in  the 
Confucian  ceremonies,  though  very  rarely  seen.  It  is  described  as  a 
hollow  cone  of  baked  clay  about  3i  inches  high,  having  a  mouth-hole 
at  the  top,  three  equal  finger-holes  on  one  side,  and  two  equal  holes  on 

the  other.  The  descriptions  available 
are  inconsistent  and  incomplete,, but 
that  given  by  Amiot  a  century  ago  is 
the  fullest.  He  reports  the  scale  as  re^ 
fa,  soL  la,  do,  re,  and  as  he  gives  a  cut 
(fig.  5),  also  the  diameters  of  holes  and 
the  external  measures,  an  approximate 
calculation  can  be  made  of  the  scale  b}^ 
the  law^s  of  resonators.  The  pitch  of 
the  fundamental  comes  out  D  above 
middle  C,  and  the  other  notes,  F,  G, 
and  A,  for  one  side;  then  starting  anew  for  the  other  side  we  get  C  and 
D,  all  within  a  quarter  of  a  semitone.    This,  it  will  be  noticed,  is  a  five- 

'  Engel,  Music  of  the  Most  Ancient  Nations,  p.  75. 

■'  History  of  Music,  II,  p.  628. 

•*  Chinese  Music,  Shanghai,  1884,  p.  82. 

^Memoires  concernant  I'histoire  .   .   .  des  Chinois,  p.  225. 


W> 


FiR.  5. 

CHINESE  EESONATOES. 
After  Aiuiot. 


432 


HK1'«»HT    (>K    NATIONAL    MTSKIM,    liKKt. 


Fig.  G. 

OLOBri.AR  WHISTLES. 
AfUT  Krolieiiiiis. 


Ill    t 


Kt»'i)  sf;ilc.  liUr  111. .St  ..r  th(>  (li.'oivlical  Chinese  scales.  The  agreement 
between  the  niuthnuMtlcal  th.-oiy  uiul  <)}).servation  is  strikino-ly  close. 
4.  All  th<>  cases  thus  far  ret'erivd  to  have  l)cen  of  prehistoric  or  very 
uiiciciit  iiistnini.'uts.  But  .some  curious  little  instruments  of  this  type 
ar»'  li^rurrd  l.y  FioluMiius'  (tig.  0)  as  ''a  splendid  parallel  l)etween  the 
rultiircs"  of  sonic  West  African  tri1)es  and  the  natives  of  New  Pom- 
crania.  The.se  are  littU»  whistles  made  out  of 
gourds  (a,  b,  d)  or  pottery  (c).  They  have  the 
mouth-hole  and  two,  three,  or  four  linger- 
hoh\s.  No  dimensions  arc  given.  Kraus,  of 
Florence,  figures  and  describes"-  a  similar 
instnunent  from  Melanesia  made  of  a  gourd 
»;  cm.  in  diameter, having  three  tinger-holes 
close  to  the  mouth-hole  (tig.  7).  The  scale  is 
stated  to  be  A,  B,  C#,  E,  F,  but  no  further 
measures  are  given.  However,  this  series  is 
easily  obtained  by  assuming  the  diameter  of 
the  mouth-hole  to  be  1.0,  of  one  hole  0.3,  and 
of  the  others  0.6;  apparently  D#  is  omitted. 
he  Fiiiscli  Collection  in  the  American  Museum  of  Natural  History, 
in  New  York  City,  there  ai-e  several  similar  gourds  of  difi'erent  sizes 
having  three  tinger-holes.  They  are  labeled  '*  Blasekugeln,"  "used 
by  women." 

.'..  In  Kiirope  there  have  lieen  many  instruments  depending  on  the 
same  general  principle  of  resonance  in  a  nearlv  closed  cavit}'  (in  dis- 
tinction from  the  open  or  closed  organ-pipe  principle), 
]»ut  not  conforming  to  the  simple  law  already  set 
forth.  Pra'torius''  in  his  famous  ]>ook  of  1618  gives 
tigures  and  descriptions  of  several  such  instruments, 
along  with  the  recorders,  flutes,  violins,  etc.,  that  one 
reads  of  more  frcHjuently;  for  instance,  he  says  the 
fagotti  are  sometimes  closed  at  the  extreme  end,  but 
have  a  side  hole:  the  Cornamuse  has  the  end  closed  and 
holes  in  the  side.  Besides  these  he descrilx's  various  in- 
strunieiits  ha\  ingstopped  bodies  on  reeds — the  rankett, 
hear  pi|)es.  etc..  and  similar  forms  on  the  organ.  These  things  have  all 
gone  out  of  use  iiloiig  with  the  other  delicate  and  weak-toned  instru- 
ments of  their  times.  To-day  musicians  demand  tones  more  powerful 
and  richer  in  harmonics  than  instruments  of  this  t^'pe  can  give,  l^ut 
a  curious  sur\ival  or  revival  of  this  earlier  type  occurred  in  the  middle 
of  this  century,  which  is  told  of  in  Groves's  Dictionary  of  Mu.sic.  A 
blind  pi'assint,  named  Picco,  gave  public  performances  in  London  on  a 

'  Her  rrxprniijf  dor  .\frikanisclu>n  Knlturen,  1898,  p.  150. 
•'  .Vnliivii)  iMT  L'.\ntr()pi)l(it:iu  e  la  ICtnolotria,  XVII,  1887,  j.p.  85-41,  fig.  5. 
Syntagma  Miisiiiiin  II.  \>\>.  44,  4S,  So. 


Fig.  7. 

(i  LOBULAR  WHISTLE. 
Aftpr  Kriiiis. 


HISTOEY    OF    MUSICAL    SCALES.  433 

flageolet  2  inches  long  and  having  only  three  holes.  By  partial!}^  or 
wholly  closing  the  end  of  the  tube  with  his  hand  he  made  use  of  the 
resonator  principle  to  lower  the  pitch  of  his  notes;  so  he  obtained  a 
compass  of  more  than  two  octaves.  The  instrument  is  similar  to 
Prtetorius's  schwdgeV  except  that  it  is  shorter,  and  the  accuracy  of 
the  notes  performed  would  depend  almost  wholly  on  the  performer. 
Later  a  traveling  troupe  appeared  in  European  cities  with  seven 
instruments  called  ocarinas.  These  are  familiar  to  us,  being  on  sale 
everywhere.  They  are  properly  resonators,  but  the  holes  are  more 
numerous  than  in  the  instruments  alread}^  considered  and  vary  widely 
in  size.  The  scale,  which  the  instruments  furnish  with  more  or  less 
precision,  is  not  dependent  on  any  simple  principle,  but  is  adjusted  l\y 
the  maker  by  varying  the  sizes  of  the  holes  so  as  to  conform  to  a  scale 
fixed  on  other  instruments. 

V.  THE  INFLUENCE   OF  THE   HAND. 

All  the  instruments  of  the  three  groups  now  discussed  are  ' '  fingered;" 
that  is,  the  acoustical  dimensions  of  the  vibrating  string  or  mass  of 
air  are  varied  as  the  plaj^er  manipulates  the  fingers  of  one  or  both 
hands.  These  instruments  therefore  involve  a  feature  not  associated 
with  drums  and  othernnstruments  of  percussion,  or  with  primitive 
harps.  Instead  of  using  the  hand  as  a  whole,  the  more  delicate 
fingers  are  utilized  separately;  so  the  simple  instrument  becomes  in  a 
peculiar  sense  a  part  of  the  player's  means  of  self-expression  and  is 
specially  responsive  to  his  own  moods,  as  many  legends  of  the  power 
of  music  testify.  But  leaving  to  the  musical  writers  such  compari- 
sons between  instruments,  it  is  important  to  the  physicist  to  recognize 
that  the  dimensions  of  the  human  hand  have  fixed  absolutely  some 
dimensions  of  these  instruments. 

The  first  thing  to  strike  one,  considering  the  hand  from  this  point  of 
view,  IS  the  fact  that  only  with  difiiculty  can  the  five  digits  be  brought 
into  line,  so  the  thumb  is  not  used  on  primitive  instruments  for  finger- 
ing, so  far  as  observed.  In  the  more  highly  developed  flutes  there 
ma}^  be  a  hole  for  it  on  the  back  side,  while  on  our  own  flutes,  clari- 
nets, etc.,  it  governs  one  or  more  kej^s.  Similarly,  the  little  finger 
does  not  readily  fall  in  line  with  the  three  longer  ones,  and,  besides,  is 
much  weaker.  The  remaining  three  fingers  on  a  hand  of  medium  size 
can  be  brought  into  a  space  of  about  1  cm.,  or  spread  to  span  perhaps 
12  cm.  (5  inches).  To  fix  one's  ideas  before  comparing  these  limits  with 
measures  on  some  actual  instruments,  it  will  be  convenient  to  recall 
that  on  piano  keyboards  the  distance  between  key-centers  an  octave 
apart  is  165  mm.  (Oy  inches),  the  same  as  on  a  spinet  of  1602;  but  on 
the  ph3^siologically  designed  Janko  kej'board,  with  the  octave  distance 

^  Syntagma  Musicum,  J).  39,  pi.  ix. 


4;U  RKl'OKT    OF   NATIONAL    MUSEUM,   1900. 

14(»iiiiii.  (."iA  incln's),  iiii  orilitiiirv  liiiiul  CUM  readily'  span  an  octaxc  and  a 
Fifth,  horausc  the  lingers  aiv  not  foivcd  into  line. 

Exsiininin*^  liist  some  string  instruments,  it  is  found  that  on  a  guitar 
of  New  York  make  (No.  55r)90,  U.S.N.M.)  the  distance  between  frets 
ranges  from  MM  to  14  mm.  The  greatest  distance  noticed  between 
fr»>ts  is  on  the  large  Siamese  Kra  Chapjh'e  (No.  27810,  U.S.N.M.), 
w  here  tliere  are  three  spaces,  respectively,  of  71,  78,  and  77  mm.  A 
siniilai'  instrument  examined  at  the  World's  Fair  held  in  Chicago  had 
the  coirespt)nding  spaces  60,  60,  and  67  mm.  The  string  lengths  to 
the  lirst  frets  were,  respectively,  878  and  740  mm.  The  smallest  dis- 
tance ol)served  between  frets  is  the  above-cited  14  mm.,  except  that 
the  Syrian  lute,  Blzug  (No.  95144,  U.S.N.M.),  has  two  spaces  of 
12  and  18  nnn.  On  most  insti'uments  the  frets  cease  when  the  limit 
of  20  to  2.")  mm.  is  reached.  It  is  obvious  that  these  and  similar  data 
for  fretted  instruments  are  not  of  much  importance  unless  one  can 
know  that  the  hand  was  not  shifted  from  one  fret  to  another. 

^^'ith  our  instruments  shifting  is  notoriously  common,  but  the  histories 
of  the  violin  report  that  two  or  three  centuries  ago  it  was  a  notable  thing 
for  a  i)layer  to  shift.  The  usual  theory  of  the  old  many-stringed  instru- 
ments, of  which  the  Aral)  lute  is  a  particularly  good  example,  required 
the  strings  to  be  tuned  in  Fourths,  and  the  strifig  lengths  were  not  too 
great  foi-  the  four  fingers  to  govern  all  the  frets  within  this  range — that 
is,  in  a  (juarter-length  of  the  string — so  a  shift  would  l)e  unnecessary. 
On  the  Aral)  lute  ^  there  were  sometimes  ten  very  unequally  spaced  frets 
in  tiiis  space,  but  for  any  one  tune  onh^  a  few  of  them  were  used,  and 
in  the  princi^jal  modes,  ^  OcJukj  and  Ra.st,  one  fret  each  for  the  index 
and  ring  fingers  sufficed  to  give  substantially  our  diatonic  scale. 

With  simple  wind  instruments  the  case  is  quite  different,  for  sev- 
eial  lingers  must  be  used  sinuiltaneously  to  cover  holes,  so  the  hand 
can  not  be  shifted.  In  the  Kiowa  flute  referred  to  above  the  uniform 
distance  between  holes  is  32  nun.;  in  the  stone  whistle  from  Mex- 
ico. 20  mm.;  in  the  four  Egyptian  flageolets  and  oboes  figured  by 
Viiloteau  (his  Plate  c  c)  the  intervals  are,  respectively,  12,  15,  15,  and 
8t;  mm.  '{'hese  distances  require  only  a  convenient  spread  of  the 
fingers.  Many  other  measures  can  readily  be  obtained  from  the 
accompanying  figures  with  their  appended  scales. 

If  the  musician  has  a  theory  demanding  that  the  holes  be  so  near 
together  or  so  fai-  apart  as  to  make  direct  fingering  inconvenient  or 
impossible,  keys  with  long  or  short  levers  are  added,  as  on  modern 
flutes  and  clarinets,  while  among  the  Romans  extra  holes  were  bored 
to  pro\id('  for  sevei-al  geiuM-a,  the  holes  not  needed  for  any  tune  being 
<'loscd  by  i)lugs  or  I'otating  rings. 

In  a  few  cast's  wind  instrumcMits  are  found  so  long  that  the  player's 

'l.iiii.1.  Tnivaux  .!.■  hi  •;■  Confrivs  .Ics  ( >riiTitalistc's,  1883,  i)p.  107-114,  or  Ellis, 
.Ii.unml  ..f  \\\v  Society  of  Arts,  XXXIII,  IsSo,  p.  502. 


HISTORY    OF    MUSICAL    SCALES.  435 

arm  is  too  short  to  reach  the  lower  end.  Then,  necessarily,  the  holes 
to  l)e  tingered  are  located  at  the  middle  or  upper  end  of  the  tube,  l)ut 
the  holes  are  so  small  that  the  pitch  of  the  resulting  notes  is  much 
lower  than  the  position  of  the  holes  would  suggest,  so  the  discrepancy 
to  the  ear  is  not  as  great  as  to  the  eye.  In  other  cases  the  length  is 
misleading,  for  the  holes  are  bored  obliqueh'  or  holes  are  bored  in  the 
tube  below  the  holes  to  be  tingered,  thereby  raising  and  adjusting  the 
pitch  of  the  lowest  note,  as  Mahillon  shows  in  the  Brussels  catalogue 
(Nos.  830,  1039,  1117,  1119,  and  1123)  and  Villoteau  shows  on  his  Plate 
c  c,  No.  1.  This  is  a  possible  explanation  of  the  superfluous  holes  in 
the  flute  on  the  statue  from  the  ruins  of  Susa  (Plate  2,  fig.  1),  if  the 
figure  be  accepted  as  archaeologically  correct.  In  modern  instru- 
ments, as  is  well  known,  the  distant  holes  are  controlled  by  covers  at 
the  ends  of  long  levers. 

The  relation  of  the  instruments  of  the  resonator  type  to  the  hand  is 
too  obvious  to  need  discussion;  the  objects  must  be  of  such  size  and 
shape  as  to  be  held  by  the  hand  or  by  two  hands  while  the  fingers  are 
manipulated,  and  the  holes  must  be  conveniently  located  and  small 
enough  to  be  closed  by  the  tips  of  the  fingers,  or  in  the  Chinese  Jiiuen 
also  by  the  thumbs. 

It  is  rather  surprising  to  see  how  little  the  thumb  is  used  in  plaj'ing 
upon  the  instruments  under  consideration.  Although  from  its  anatom- 
ical structure  the  thumb  has  a  peculiar  independence  in  its  movements, 
yet  most  of  its  services  are  rendered  by  cooperation  with  the  other  fin- 
gers; and  the  natural  training  of  these,  as  in  grasping,  sewing,  weav- 
ing, or  the  most  delicate  savage  industries,  appears  likewise  to  call  for 
their  cooperation,  not  for  independent  action.  It  is  onh'  in  plajdng 
instruments  like  the  lyre  and  harp  (whose  tuning  depends  on  princi- 
ples outside  the  instrument,  and  so  they  do  not  belong  to  the  present 
discussion)  that  one  sees  a  grasping  action  requiring  two  or  more  fin- 
gers at  once.  But  in  the  guitars,  flutes,  etc.,  under  consideration,  the 
thumb  is  constantly  occupied  in  merely  supporting  the  instrument,  so 
any  variation  in  the  pitch  of  the  sound  can  come  only  as  the  other  fin- 
gers become  independent  in  action.  When  we  remember  how  difficult 
it  is  for  a  civilized  piano-player  or  typewriter  to-day  to  acquire  a  sat- 
isfactory independence  in  movement  of  all  the  fingers,  especially  of 
the  third  and  fourth,  and  recall  that  the  early  instruction-books  for 
the  harpsichord  required  the  use  of  but  two  fingers  on  each  hand,  we 
shall  have  a  higher  respect  for  the  technique  of  primitive  musicians, 
and  shall  not  wonder  that  primitive  wind  instruments  have  so  few 
holes.  Presumably  the  index  finger  first  gained  independence,  and  then 
it  marked  a  long  advance  when  two  fingers  could  act  independently  of 
one  another.  So  the  four-hole  flute  or  resonator,  requiring  the  action 
of  two  fingers  from  each  hand,  and  giving  a  scale  of  five  tones,  is  a 
monument  commemorating  an  important  stage  both  in  the  development 
of  the  hand  and  in  the  extension  of  musical  resources. 


43<) 


RErOKT    OF    NATIONAL    MUSEUM,   190(>. 


VI.  COMPOSITE   INSTRUMENTS. 

Eju-li  of  the  instruments  thus  far  examined  is  capable  of  furnishing 
s('\  fr:il  not«'s  of  approximately  constant  pitch,  but  the  general  princi- 
))!(•  before  us  may  b(\  eml)odied  in  composite  instruments,  where  each 
note  has  its  ow  n  vibrating  l)ody,  thus 

1.  Various  forms  of  harps  and  dulcimers  show  strings  of  regularly 
decreasing  h'ligth;  here,  of  course,  difference  of  tension  may  nullify 
the  scale  due  to  the  lengths.     One  form  is  shown  on  Plate  8. 

2.  Pan's  ])ipes  are  soiuetimes  seen  with  regularh'  decreasing  lengths; 
it  is  true  that  this  regularity  is  not  very  common,  but  it  is  the  only 

principle  of  scale  )>uilding  (except 
the  Chinese  cycle  of  fifths)  yet  recog- 
nizable in  these  priuiitive  instru- 
ments.    (Plate  9.) 

3.  Instruments  of  the  bar  tA'pe  are 
found  fre((uently  in  our  orchestras 
and  ])ands  under  various  names,  as 
•I'ylophone;  they  are  familiar  in 
children's  toys  and  are  widely  dis- 
tri])uted  in  savage  and  half -civilized 
lands  under  the  names  of  rnarlmha^ 
hdhifong^  liarmonicon^  etc.  (Plate 
8  aiid  fig.  8.)  The  law  of  the  uni- 
form bar  is  that  the  frequencies  of 
vibration  of  a  series  of  bars  of  the 
same  material  are  proportional  to 
the  quotients  of  the  thickness  divid- 
ed l)y  the  square  of  the  length;  the 
breadth  is  immaterial  if  it  is  uni- 
form. So  if  one  takes  a  series  of 
miiforiii  bius  of  tlie  .sun(;  thickness  and  regularly  decreasing  length  he 
may  ottlaiii  a  series  of  ascending  notes.  Thus,  let  the  first  bar  be  24 
units  long  (tor  example  24  cm.),  the  successive  bars  decreasing  by  one 
unit;  the  eighth  bar  will  be  IT  units  long,  and  the  fifteenth  l)ar  10 
uiiit->:  the  series  of  fi'(>(|ueiu'ies  would  then  be  as  the  reciprocals  of  the 
s(|uares  of  'lA,  23,  etc.,  so  giving  to  the  ear  a  series  of  increasing  inter- 
vals; with  these  jji'oportions  bar  No.  8  would  give  the  Octave  of  the 
first,  but  b;ir  No.  !.")  would  give  the  Twelfth  of  bar  No.  8.  The  sim- 
plicity of  the  rule,  however,  frecpiently  disappears,  either  because  of 
vaiiations  in  the  thickn(>ss,  as  when  a  savage  splits  a  bamboo  stem  and 
then  cuts  his  bars  so  that  the  shorter  ones  are  also  thinner,  or  because 
of  the  attachment  of  lumps  of  wax  or  clay  to  the  bars  to  tune  them  to 
some  other  in-^truuient;  or  because  of  the  hollowing  of  the  center,  as 
is  done  by  uiodein  .Japanese;  so  at  pi'esiMit  one  can  not  atfirm  that  this 


v^ 


Fig.  H. 

.\YI.()1'H(>NE.S. 
After  Kraiis. 


HISTOEY    OF   MUSICAL    SCALES.  487 

theoretical  principle  of  scale  determination  is  certainly  and  consciously 
embodied  in  any  instrument  anywhere;  but  some  instruments  in  the 
National  Museum  and  some  drawings  in  books  make  the  assumption 
seem  plausil^le,  that  the  primitive  type  of  this  instrument  is  a  seric^s  of 
bars,  supported  at  points  about  one-tifth  or  one-fourth  of  their  lenglh 
from  their  ends,  and  decreasing-  in  length  by  equal  linear  amounts. 

It  is  evident  that  tliese  composite  instruments  are  of  minor  impor- 
tance in  this  study;  but  in  the  light  of  the  theoretical  laws  here 
suggested  perhaps  travelers  may  learn  something  of  the  intention  of 
a  savage  who  cuts  his  Pandean  pipes  or  bars  to  form  a  musical 
instrument. 

VIL  CONCLUSIONS. 

There  have  now  been  considered  all  the  types  of  instruments  in 
which  several  notes  of  different  pitch  are  produced  from  the  same 
vibrating  body — whether  string,  column  of  air,  or  mass  of  air. 

(1(1.)  There  have  been  found  examples  from  various  parts  of  the  world 
of  the  intentional  location  of  the  stopping  points  of  a  vibrating  string  at 
equal  linear  distances;  since  with  all  stringed  instruments  the  linger- 
ing will  cause  a  slight  increase  of  tension,  the  equivalent  length  of  the 
string  is  less  than  the  actual  length. 

(IJ.)  There  have  been  found  numerous  examples  of  wind  instru- 
ments pierced  with  holes  in  one  or  two  groups  spaced  at  equal  linear 
distances;  since  these  holes  are  never  sufficiently  large  to  allow  the  air 
to  flow  through  them  with  perfect  freedom  (unless  in  some  Chinese 
flutes)  the  equivalent  length  of  the  vibrating  column  of  air  is  greater 
than  the  actual  distance  from  the  mouthpiece  to  the  hole. 

(2.)  There  have  been  found  instruments  of  the  Marimba  type  with 
bars  of  regularly  decreasing  lengths. 

(3.)  There  have  been  found  many  forms  of  instruments  of  the  resona- 
tor type  embod3dng  a  series  of  equal  and  similarly-located  holes;  in 
these,  thickening  the  wall  is  equivalent  acoustically  to  making  the  holes 
smaller;  while  locating  the  hole  nearer  the  point  where  the  vibrating 
air  has  its  maximum  change  of  density  is  equivalent  to  enlarging  the 
hole. 

Three  simple  laws  give  to  the  first  approximation  the  scales  of  these 
several  instruments,  namely: 

(1)  The  law  of  inverse  lengths. 

(2)  The  law  of  inverse  squares  of  lengths. 

(3)  The  law  of  the  square  roots  of  a  series  of  numbers  propor- 

tional to  sums  of  diameters. 
The  first  and  second  laws  give  scales  whose  intervals  increase  as 
the  j)itch  rises;  scales  based  on  the  third  law  have  decreasing  inter- 
vals.    Some  results  are  shown  in  a  table  in  the  appendix  and  graphi- 
cally in  Plate  10. 


438  RKTOHT    OF    NATIONAL    MUSEUM,  1900. 

Fioin  those  it  is  evident  that  whiehever  type  of  iMstruinont  one  ina}^ 
take,  there  will  he  some  intervals  that  very  closely  agree  with  inter- 
vals of  our  familiar  scale.  In  a  few  cases  this  comes  about  because 
our  scale  is  ])rinci|)aily  derived  from  the  Greek  theorists,  who  l)ased 
their  scales  on  proportional  string-lengtlis;  so,  if  the  unit  of  equal 
distance  on  a  simple  guitar  chances  to  be  an  aliquot  part  of  the  length 
of  the  string  from  the  ])ridge  to  the  nut,  some  of  the  resulting  notes 
will  Ixdongto  our  scale.  (However,  the  divisor  must  not  have  a  prime 
factor  greater  than  live.)  But  whatever  the  instrument,  on  any  doc- 
trine of  cliances,  there  will  l)e  some  ai)proxinjate  coincidences;  and 
these  coincidences,  as  judged  by  the  ear,  will  l)e  found  nuich  closer  and 
moi"e  luuiierous  than  when  judged  mathematically  or  graphically;  for 
the  training  of  modern  musicians,  as  has  often  ])een  recognized,  not 
only  allows  ])ut  compels  them  to  ignore  deviations  from  their  standard 
scale  deviations  amounting  sometimes  to  more  than  half  a  semitone. 
So  one  is  forced  to  conclude  that  the  recognition,  even  l)v  a  nnisically 
educated  ear.  of  a  series  of  notes  as  agreeing  substantially  with  our 
diatonic  scale  or  with  any  other  known  scale,  does  not  afford  any  ade- 
i{uat«'  ground  for  judging  of  the  principles  underlying  the  series:  in 
fact,  the  failure  to  note  the  deviation  may  prevent  the  recognition  of 
the  underlying  principle. 

The  type  Costa  Kican  four-hole  whistle  is  the  most  striking  example 
of  a  series  agreeing  closely  with  notes  of  our  scale,  j^et  based  on  an 
absolutely  different  principle:  for  the  mean  computed  deviation  from 
the  piano  intervals  is  only  one-eighth  of  a  semitone. 

Further,  the  whole  discussion  makes  it  evident  that  the  people  who 
made  and  used  these  instruments,  or  any  single  type  of  them,  had  not 
that  idea  of  a  scale  which  underlies  all  our  thinking  on  the  subject, 
namely:  A  series  either  of  tones  or  of  intervals  I'ecognized  as  a  stand- 
aid,  independent  of  any  particular  instrument,  but  to  which  every 
instrument  nmst  conform.  Modern  Europeans  for  the  sake  of  har- 
mony have  nearly  banished  all  scales  but  on(\  and  seldom  know  ))y  what 
rules  the  instruments  are  tuned  to  furnish  this.  But  for  these  people 
the  instrument  is  the  primaiT  thing,  and  to  it  the  rule  is  applied,  while 
the  scale  is  a  result,  or  a  secondai-y  thing:  and  the  same  rule  applied  u 
hundred  tinirs  may  possil)ly  give  a  hundred  different  scales.  Natur- 
ally one  does  not  expect  to  find  nuich  concerted  music  among  people 
in  this  stage  of  de\elopment. 

'I'he  \arious  rules  discussed  aboNc  may  he  united  in  a  generic  one, 
namely: 

7//'  jirninini  juinc'ipli  in  tin  nnik'nuj  of  iinmical  itixfruiiients  that 
i/i,hl  II  scale  ix  the  repetition  of  >'leiiumts  similar  to  the  eye;  the  size ^ 
innntur^  and  location  of  these  dements  heiny  dependent  on  the  size  of 
th.   Iiiind  and  the  di(/itnl  e,r/)ertness  oft/ie  performer. 

'riii>  piinciplr  shows  itself  in  the  occasional  eipial  spaces  on  the  neck 


HISTORY    OF    MUSICAL    SCALES.  439 

or  table  of  ti  .stringed  iiistruuient,  and  conspicuously  in  the  series  of 
holes  on  flutes  and  primitiAe  oboes,  while  a  sense  of  balance  and  sym- 
metry added  to  the  repetition  appears  in  the  two  groups  of  holes  on  the 
flutes,  etc.,  and  especially  in  the  resonators,  and  appears  in  a  different 
wa}'  in  the  trapezoidal  forms  of  dulcimers.  Pan's  pipes,  and  marimbas. 
The  pitch-determining  elements  are  therefore  primarily  decorative. 
In  fact  no  one  can  examine  any  collection  of  primitive  wind  instru- 
ments^ or  drawings  of  them,  without  being  struck  by  the  way  in  which 
the  holes  often  cooperate  in  the  decoration;  while  the}'  are  not  found 
interfering  with  the  artistic  design  (see  fig.  'S,  page  430;  Plate  2,  figs.  1 
and  2;  Plate  3,  fig.  .2). 

Simple  decoration  involving  only  repetition  and  synnnotrical  placing 
or  grouping  of  similar  parts  is  not  only  found  among  living  primitive 
peoples  everywhere  that  musical  instruments  embodying  a  scale  can 
be  found,  but  is  prehistoric.  The  prehistoric  flutes  are  believed  to 
come  from  the  neolithic  age,  and  the  pottery  from  this  age  shows  a 
multitude  of  geometrical  designs,  some  of  which  are  collected  in  Wil- 
son's Plates  19  and  20.  The  paleolithic  age  has  furnished  few  geomet- 
rical designs  and  no  flutes  or  many-holed  resonators.  In  applying  such 
decoration  to  the  hollow  bones  of  animals  or  human  enemies,  to  the  hol- 
low reeds  that  Lucretius  says  whistle  in  the  wind,  or  to  gourds  and  sim- 
ple pottery,  nothing  can  be  more  natural  than  sometimes  to  perforate 
the  walls  and  to  get  a  several-toned  musical  instrument  as  the  result.  So 
although  no  conclusions  regarding  the  mental  operations  of  prehistoric 
man  can  be  absolutely  certain,  one  feels  a  strong  conviction  that,  as 
with  immature  minds  among  us,  art  appealed  first  to  the  eye  and  later 
to  the  ear;  that  beauty  of  material  form  incidentalh"  furnished  series 
of  sounds  that  could  be  repeated,  and  could  give  to  the  ear  and  the 
mind  the  idea  of  the  definite  leaps  or  steps  that  Aristoxenus,  countless 
ages  afterward,  called  the  characteristic  of  music.  (Of  course  rhythm 
in  movement  and  in  sound  are  independent  of  the  structure  of  an 
instrument.)  Any  influence  that  may  have  been  exerted  on  the  estab- 
lishment of  scales  b}"  the  songs  of  birds,  b}"  the  recognition  of  over- 
tones in  the  sounds  of  the  human  voice,  or  by  the  production  of  har- 
monics on  the  horn  must  have  been  limited  and  trivial.  The  principle 
here  presented  is  at  any  rate  a  'vera  causa,  and  explains  facts  hitherto 
unexplained;  further,  (1)  it  is  extremeW  simple  both  in  theory  and 
practice;  (2)  it  is  flexible,  allowing  of  multifarious  results  in  prac- 
tice; (3)  it  is  suggested  by  prehistoric  instruments,  supported  b}'  the 
instruments  of  many  living  primitive  peoples  and  repeatedly  con- 
firmed by  its  survival  in  several  instruments  of  peoples  in  an  advanced 
stage  of  musical  culture. 

It  only  remains  to  add,  in  order  to  prevent  misunderstanding,  that 
the  principle  here  set  forth  never  appears  as  the  dominating  one  among 
peoples  who  are  known  to  have  had  a  theory  of  the  scale.     The  Greek 


440  REPORT  OF  NATIONAL  MUSEUM,  190(). 

theoretical  scales,  diatonic  and  noiidiatonic,  are  doubtless  its  direct 
descendants,  thouirh  at  present  it  is  not  known  what  the  influence  was 
that  so. transformed  them  and  made  them  depend  on  ratios,  not  on  dif- 
ference of  lengths.  Possibly  the  theor}'  of  numbers  bewitched  musi- 
cians then  as  it  has  sometimes  since,  though  tiie  converse  speculation 
is  a  plausi])le  one — that  the  recognized  musical  ratios  gave  a  mj^stical 
meaning  to  numl)ers.  It  is  curious  to  note  that  Aristoxenus  had  some- 
how got  fai-  enough  to  complain  that  flutes  distort  most  of  the  inter- 
vals (p.  4"2,  Mb.),  and  if  his  lost  treatise  on  boring  flutes  should,  be 
found  it  might  throw  light  on  this  history.  The  Arab  ''step  by  step" 
metliod  is  a})i)arently  a  kite  descendant  of  the  equal  linear  divisions, 
appearing  after  men  had  learned  to  recognize  the  equality  of  intervals 
as  well  as  of  sjjaces.  But  the  Chinese  cycle  of  flf ths  must  be  explained 
and  determined  on  entirely  difterent  physical  principles,  and  the  vari- 
ous P^uropean  scales  as  defined  b}^  theorists  or  rendered  by  the  best 
violinists  or  fixed  by  good  tuners,  when  properlj^  examined,  reveal 
elements  as  diverse  as  the  elements  of  our  language  or  our  population. 
The  principle  in  question  is  therefore  presented  only  as  the  simplest, 
earliest,  and  most  primitive  principle  of  scale-building. 


HISTORY    OF    MUSICAL    SCALES.  441 


APPENDIX. 

The  laws  briefly  stated  on  page  437  for  the  several  kinds  of  instruments  discussed 
in  the  paper  may  be  expressed  more  accurately  by  the  following  fonnuke: 

Let  N  =  number  of  complete  vibrations  per  second. 
/  =  length  of  string  or  c<dumn  of  air  or  bar. 

a  =  diameter  of  mouth-hole  of  resonator,  corrected  for  thickness  of  wall. 
f)  =  diameter  of  finger-holes  of  resonator,  corrected  for  thickness  of  wall. 
n  —  number  of  finger-holes  opened  on  resonator. 
t  =  thickness  of  bar. 
K  =  constant,  depending  on  material  and  units  of  measurement. 
Assuming  centimeter-gram-second  units  and  ordinary  temperatures, 

K"  =  \/Tension  in  dynes  -f-  mass  in  grams  per  cm.  =  velocity  ;  e.  g.,  in  piano 
strings  17,000  to  40,000  cm. -sec;  in  violin  strings  from  13,000  for  the 
covered  string  to  43,000  for  the  gut  E-string;  in  weak  primitive  instru- 
ments probably  much  less. 
K"  =  34,000  cm.-sec,  the  velocity  of  sound  in  air. 
K"'  =  520,000  cm.-sec.  for  iron  bars;  340,000  to  520,000  for  wood  bars  supported 

as  usual  in  a  xylophone. 
K'^  =  5,500. 

Then,  corresponding  with  the  brief  laws, 

( la)  For  strings:  N  =  ,^j-^=  ..^ — ^—  v  tension  ---  linear  density- 

{lb)  Forcolumnsof  air:  N=  27:17= — Tli:^- 

(2)  For  bars:  N  =  K"'  4=  340,000  to  520,000  * 

(3)  For  resonators:  N  =  K"'  ^^^'H^»m  of  h  ^5^00^      lf-^^^b\ 

V  volume  v  volume'Y     v  «y 

These  constants  are  sufficiently  accurate  for  the  general  purposes  of  the  anthro- 
pologist and  musician.  But  the  results  should  be  expressed  in  musical  terms.  The 
French  standard  pitch,  now  adopted  by  the  Piano  Makers'  Association,  gives 
A  =  435  d.  v.,  or  C  =  258.7  d.  v.,  and  the  ratio  for'any  interval  of  p  piano  semitones 
is  2 1?.  In  most  cases  it  is  much  more  convenient  to  have  intervals  than  ratios;  and 
incomparably  the  most  convenient  unit  of  intervals  is  the  piano  semitone,  of  which 
12  by  definition  make  an  octave;  these  can  readily  be  grouped  by  anyone  with  slight 
musical  knowledge  into  larger  intervals.  Thirds,  etc.,  and  the  musical  value  of  any 
whole  numlier  of  them  can  instantly  be  found  on  a  well-tuned  piano. 

Since  the  reduction  of  ratios  to  intervals  can  not  ordinarily  be  done  without 
logarithms,  a  short  table  has  l^een  calculated  and  is  appended  by  the  use  of  which 
the  reduction  may  be  done  by  inspection  in  most  practical  cases.  This  table  gives 
the  logarithm  of  every  whole  number  from  1  to  40,  and  the  product  of  these  by  40, 
less  one  three-hundredth,  together  with  the  successive  differences;  these  are  in 
semitones;  for  the  factor  is  so  chosen  that  when  the  logarithm  of  the  ratio  2:1  is 
multiplied  by  it  the  product  will  be  12,  which  is  the  number  of  semitones  corre- 
sponding to  the  ratio  of  the  octave.  Much  more  elaborate  tables,  but  without  the 
column  of  differences,  have  been  published  by  Prony  and  by  Ellis.  In  using  the 
tah>le  it  is  well  to  remember  that  the  average  uncertainty  in  pitch  of  public  per- 
formers in  Berlin  was  found  to  be  about  one-tenth  of  a  semitone. 

NAT   MUS   lyUU 31 


442  BEPORT    OF    NATIONAL    MUSEUM,  1900. 

( ,1,.,  „<„  „(  tlR.  tal.l.'   liri'l  the  successive  intervals  of  tVie  scale  of 
As  il  ustrati.uis  of  the  Ubc  i)l  tlit  i.inn    ii 
the  HiM.h.  .»..■,  the  strins  heing  stop,.e.l  s,.cces»,vely  at  36  3o  34,  .  .  .7 
ponclh.,  .liffcToces  in  column  4  o,  the    f  ''^  -^  "f^^^^f,; '  To'  omplet;  the 

^;ro"37s;^.?iru:5^^^^^^ 

•";,r- u::,»2t  ;"thet;urelt:i\wttir.t,^ 

,r.:.alL':r;4  butV  tlnal  results  ate  to  b^^^^^^^^^ 

1,,..  type  resonator,  calling  the  equu^  ent  -f^  "^^f  ^f^'^,"  j,„,;,„„  be  I.O. 
,in...r  holes  is  0.«  (nu;re  --»-'f J'^';^^  ", ;J'    ,  *^^^^^^^  nnn.bers from  cohunn 

;-:ith;t^;ie.  ;«:*£  :S«"r;:;raua  aaa  the^uotlents  to  the  funda^eutal 
pitch.     The  reaults  are  as  follows: 


10 
16 
22 

28 
34 


E.  S. 

39.86 

48.00 

53.51 

57.69 

61.05 


8.14 
13.65 
17.83 
21.19 


4.07 

6.83 

8.92 

10.60 


A  +  .07  E.  S. 
C-.17. 
I)  -  .08.      ■ 
E  -  .40. 


Tf  n  fio  ha,l  been  taken  the  results  wonl<l  have  been  slightly  higher  in  pitch 

be  doubled  instead  of  halved.     Tim.,  ^ut^l  a  .enes  o i^  ^    ^^_^.^^  .^ 

24,  23,  etc.,  to  17,  the  compass  will  be  2  X  (.55.02  -  4y.uo  j 
practically  an  octave,  as  stated  on  page  436. 


Table  for  computing  musical  mtcrvaU. 


E.  S. 


Dif. 


52.70 

0.84 

53.51 

.81 

54.28 

.77 

55.02 

.74 

55.73 

.71 

56.41 

.68 

57.06 

.65 

57.69 

.63 

.58.30 

.61 

58.88 

.58 

.59.45 

.57 

60.00 

..55 

60. 63 

.53 

61.05 

.52 

61.55 

..50 

62.04 

.49 

62.52 

.48 

62. 08 

.46 

63.43 

.45 

63. 86 

.43 

EXPLANATION    OF    PLATE    1. 
STRINCIED   INSTRUMENTS. 

Kiu'.  I.  Small  Turkish  Tamboura. 

(Cat.  No.  95312,  U.  S.  N.  M.) 

Fig.  2.  Medium  Colascioni  (Italian). 

(Cat.  No.  95307,  U.  S.  K.  M.) 
X,)TK.— The  scale  shown  on  this  and  most  of  the  foUowhig  plates  is  20  centimeters 
long. 

444 


Report  of  U.  S.  National  Museum,   1900.— Wead. 


Plate  1 


Stringed  Instruments. 


EXPLANATION  OF  PLATE  2. 

FLUTES  WITH  EQUAL-SPACED  nOLES,  TYPE  A. 

Fig.  1.  Pipe  from  Susa.     Engel,  Music  of  the  Most  Ancient  Nations,  p.  77. 

Fig.  2.  Bone  Flute,  about  6  inches  long,   disinterred  at  Truxillo,   Peru.     British 

Museum.     Engel,  Musical  Instruments,  p.  64. 
Figs.  3,  4.  Aztec  Pipes,  called  by  Mexicans  pito;  usual  form;  scale,  a,  b,  c#,  e,  f#. 

Engel,  ]\Iusi('al  Instruments,  p.  62. 
Fig.  5.  Aztec  Pipe;  unusual  form.     Engel,  Musical  Instruments,  p.  62. 

446 


Report  of  U.  S.  National  Museum.  1900.— Wead. 


Plate  2. 


Flutes  with  equal-spaced  Holes. 


J 


EXPLANATION     OF     PLATE    3. 

FLUTES    WITH    EtiUAL-SPACED  HOLKS,    TYI'K    A. 

Ki<r.  1.  Double  FL.\(iEOLET.     Mexico. 

(Cat.  No.  197173,  F.  S.  N.  M.     Report  18%,  fig.  2.'K)b.) 

Fi<i.  1'.  Aztec  Flageolet  {pita).     Mexico. 

(Cat.  No.  172819,  U.  S.  N.  M.     Report  1890,  fig.  252.) 

Fig.  M.  Stone  Flageolet.     Mexico. 
(Cat.  No.  98948,  U.  S.  N.  M.) 

Fig.  4.  BoxE  Fl.\geolet.     Costa  Rica. 

(Cut.  No.  18108,  U.  S.  N.  M.    Report  1890,  fig.  273.) 

Fig.  5.  Bone  Flageolet.     Amazon. 
(Cat.  No.  5719,  U.  S.  N.  M.) 

Fig.  <>.   15.VMBOO  Whistle.     Thibet. 

(Cut.  No.  ]r)716.5a,  U.  8.  N.  M.     Rei)ort  1890,  plate  09.) 

Fig.  7.    Ba.mhoo  AVhistle.     Thibet. 

(Cut.  .No.  l(;7105b,  U.  S.  N.  M.     Report  18%,  plate  09.) 

l"Mg.  H.  Shephekd's  Pipe,  with  reed.     Arabia. 

(Cut.  No.  93.5.5.5,  U.  S.  N.  M.) 

Fig.  it.    lIoKX  (SoUlolorri).     Finland. 
(Cut.  No.  9.5C).80,  U.  S.  N.  M.) 

448 


Report  of  U.  S.  National  Museum,  1900.— Wead. 


Plate  3. 


Flutes  with  equal-spaced  Holes. 


EXPLANATION    OF    PLATE    4. 

FLUTES    WITH    EQUAL-SPACED    HOLES,  TYPE   A. 

Fip.  1.  PiHECT  Flute.     Peru. 

(Cat.  No.  959W,  U.  S.  N.  M.) 

Fig.  L*.  Flxtte  or  Flageolet.     Kiowa  Indians. 
(Cat.  No.  1.53584,  U.  S.  N.  M.) 

Fig.  3.  Flute  or  Flageolet.     Mohave  Indians. 

(Cat.  No.  107535,  U.  S.  N.  M.) 
Fig.  4.  Flute  oh  Flageolet.     Dakota  Indians. 

(Cat.  No.  23724,  V.  S.  N.  M.) 

Fig.  o.  Transverse  Flute  (Ti-tzu).     China. 

(Cat.  No.  13044G,  U.  S.  N.  M.) 
Fig.  (i.  Transverse  Flute  {Komn  Fuye).     Japan. 

(Cat.  No.  93205,  U.  S.  N.  M.) 

Fig.  7.  Oboe  {Pee  Chawar).     Siam. 
(Cat.  No.  27313,  U.  S.  N.  M.) 

Fig.  8.  Flageolet  {Sopilka).     Little  Russia. 
(Cat.  No.  96466,  U.  S.  N.  M.) 

Fig.  9.  Double  Flageolet.    Thibet. 

(Cat.  No.  95816,  U.  S.  N.  M.) 
450 


Report  of  U.  S.  National  Museum,  1900.— Wead. 


Plate  4. 


Flutes  with  equal-spaced  Holes. 


m 


EXPLANATIONOFPLATE5. 

FLUTES    W  ITII    HOLES    IN    TWO    CiROUPS,  TYPE    B. 

From  Pra'torius's  Syntatrnia  ]Musicuin  of  1G18,  to  show  tinger-holes  jrrouped  in  two  sets. 
452  ■ 


Report  of  U.  S.  National  Museum,  1900 — Wead. 


Plate  b. 


geuwiK'.*'*i'/-.tm%uiikW'^^'»wy.'A!ii»^ 


CHStrms 


T.Jli\\i\i\iil 


Flutes  with  Holes  in  Two  Groups. 


EXPLANATION     OF    PLATE    6. 

FLUTES    WITH    HOLES    IN   TWO   GROUPS,  TYPE   B. 


Fig.  L  Flageolet  (Sording).     Java. 
(Cat.  No.  95669,  U.  S.  N.  M.) 

Fig.  2.  Dikect  Flute.     Ceylon. 

(Cat.  No.  95727,  U.  S.  N.  M.) 

Fig.  3.  Dikect  Flute  {Manjairnh) .     Syria. 
(Cat.  No.  951.50,  U.  S.  N.  M.) 

Fig.  4.  German  D  Flute.     New  York. 

(Cat.  No.  55624,  U.  S.  N.  M.) 

Fig.  5.  Flageolet  {Souling).     Java. 
(Cat.  No.  95666,  U.  S.  N.  M.) 

Fig.  f^.  Transverse  Flute  {Muruli).     Bengal. 
(Cat.  No.  92707,  U.  S.  N.  M.) 

Fig.  7.  Transverse  Flute.     Manila. 
(Cat.  No.  9.5061,  U.  S.  N.  U.) 

Fig.  8.  Transverse  Flute.     .Manila. 

(Cat.  No.  95060,  U.  S.  N.  M.) 
454 


Report  of  U.  S.  National  Museum,  1900. — Wead. 


Plate  6. 


Flutes  with  Holes  in  Two  Groups. 


EXPLANATION     OF     PLATE    7. 

CENTKAL    AMEKICAN    KESONATUKS    OK    \\IIISTI>ES. 

Fiij.  \.   C'osTA  Rica. 

(Report  U.  S.  Nat.  Mus.,  1896,  p.  til".    Scale:  f,  a,  c,  d,  (\    Cat.  No.  59970,  IT.  S.  N.  M.) 

Fig.  2.  Costa  Kka. 

(Report  U.  S.  Nat.  Mus.,  1896,  i\ii.  2t«.     Scale:  <1,  c,  fjt,  k,  a.     Cat.  No. 59969,  U.  S.  N.  M.) 

Fig.  8.  C'OSTA  Rica. 

(Report  IT.  S.  Nat.  Mus.,  1896,  flg.  262.     Scale:  gf,  bb,  h,  c,  dt",  d,  eb,  e,  f.     Cat.  No.  28952,  c . 

N.M.) 

Fig.  4.  CosT.v  Rica. 

(Report  U.S.  Nat. Mus.,189(i, tig. 269.     Scale:  f,K,a,bl',  e.     Cat.  No. 28956,  U.S.N.  M.) 

Fig.  5.  Costa  Rica. 

(Report  U.  S.  Nat.  Mus.,  1896,  i>.  617.    Scale:  db,  f ,  gb ,  ab,  bf .    Cat.  No.  dOO-l.'),  I'.  S.  N.  M. ) 

Fig.  (i.  Pana.ma,  Chikiqui. 

(Report  U.  S.  Nat.  Mus.,  1896,  figs.  301--5.     Holmes,  Report  Bureau  Klhiiol.,  1884-5,  figs. 
245-246.     Scale:  end  closed,  f,  g,  ab,  bb;  open,  f#,  g?:,  a#,  b.     Cat.  No.  109682,  U.  S.  N.  M.) 

Fig.  7.  (!osTA  Rica. 

(Report  U.S. Nat. Mus.,  1896, p. 614.     Scale:  gl>,bt>,cb,dl>,eb.    Cat.  No. 28954,  U.S.  N.  M.) 

Fig.  s.  Costa  Rica. 

(Report  U.  S.  Nat.  Mus.,  1896,  lig.  270.   Scale:  ab,  bb,  b,  e,  db,  d,  eb.    Cat.  No.  (i423,  U.  S.  N.  M.) 
456 


Report  of  U.  S.  National  Museum,  1900.— Wea 


Plate  7. 


Central  American  Resonators,  or  Whistles. 


NAT  MUS  1900 32 


EXPLANATION    OF    PLATE    8. 

COM I'OSITE    INSTKIIMENTS. 

V\".  1.  Pan's  Pipes.     Cairo,  Egypt. 

(Cat.No.94()53,lT:S.N.M.) 

Fig.  2.  Kaxtele.     Finland. 

(Cat.N(..95G91,U.S.N.M.) 
V\ii.  :i   MuKKiN.     Japan.     Two  bai^  turned  edfrewis^e  to  Au>w  their  form. 
(Cat.  Ko.  96841,  U.  S.  N.  M.) 

The  jiaper  scale  is  20  centimeters  long. 
458 


Report  of  U.  S.  National  Museum,  1900.— Wead. 


Plate  8. 


Composite  Instruments. 


EXPLANATION    OF    PLATE    9. 

pan's  pipes. 

Kijr.  1.  Pan's  Pipes  (/S'«/rt#r).     Egypt. 
( Cat.  No.  94633,  U.  S.  N.  M.) 

Fig.  2.  I'an's  Pipes.     Fiji  Archipelago. 

(Report  r.  S.  Nat.  Mus. ,  1896,  p.  5.59;  Cat.  No.  'iSOl'i,  U.  S.  N.  M. ) 

Fig.  ;i  Pax's  Pipes  [Hiiayra  Puhura) .     Peru,  from  an  ancient  grave. 
(Cat.  No.  130869,  U.  S.  N.  M.) 

Am 


Report  of  U,  S,  National  Museum,  1900.— Wead. 


Plate  9. 


PAN'S  Pipes. 


EXPLANATION    OF    PLATE    10. 

SCALES    GIVEN    RY    RESONATORS. 

The  construction  of  this  chart  has  been  explained  in  the  appendix.  To  use  it,  find 
in  the  base  line  the  number  which  expresses  the  radius  of  the  finger  holes,  that 
of  the  mouth  hole  being  considered  1.0,  and  erect  a  perpendicailar  therefrom; 
the  heights  of  the  points  of  intersection  with  the  successive  curves,  measured  on 
the  left-hand  scale,  give  the  pitch  of  the  successive  notes  produced  as  the  holes 
1,  2,  3,  etc.,  are  opened,  expressed  in  equally  tempered  semitones,  E.  8.  The 
dotted  line  corresponds  to  the  position  on  the  chart  of  the  type  resonator.  The 
chart  shows  clearly  how  the  successive  intervals  become  smaller  as  the  number  of 
open  holes  increases,  and  how  the  total  compass  is  small  if  the  finger  holes  are 
relatively  small. 

Use  may  be  made  of  the  chart  for  many  ready  calculations  of  intervals  other  than 
those  due  to  equal  differences,  and  by  doubling  the  readings  in  E.  8.  the  result 
may  be  applied  to  string  ratios;  e.  g.,  find  the  interval  corresponding  to  the  ratio 
5 :  4,  or  1+0.25;  the  chart  gives  directly  1.9;  the  double  of  which  is  3.8  E.  8.  The 
table  ni  the  appendix  gives  more  accurately  3.86  E.  S.,  showing  that  the  just 
Third  is  0. 14  E.  S.  flatter  than  the  piano  Third. 

462 


Report  of  U.  S.  National  Museum,  1900.— Wead. 


Plate  10. 


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Scales  given  by  Resonators. 


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